A note on modular Terwilliger algebras of association schemes

TL;DR

This paper investigates the structure of modular Terwilliger algebras of association schemes, defining the primary module, classifying composition factors, and characterizing schemes based on algebra properties.

Contribution

It introduces the primary module concept for modular Terwilliger algebras and characterizes $p'$-valenced schemes through algebraic properties.

Findings

Determined all composition factors of the primary module.

Characterized $p'$-valenced schemes via modular Terwilliger algebra properties.

Provided corollaries related to the structure of these algebras.

Abstract

Let $p$ denote a prime number. In this note, we focus on the modular Terwilliger algebras of association schemes defined in [3]. We define the primary module of a modular Terwilliger algebra of an association scheme and determine all its composition factors up to isomorphism. We then characterize the $p'$-valenced association schemes by some properties of their modular Terwilliger algebras. The corollaries about the modular Terwilliger algebras of association schemes are given.